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Class 12 Mathematics

Chapter 2 Solutions — Inverse Trigonometric Functions

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Overview

Step-by-step NCERT solutions for Inverse Trigonometric Functions (Chapter 2, CBSE Class 12 Mathematics) — every question and answer worked out in full, not just the final result. You can also read the Inverse Trigonometric Functions textbook chapter.

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What these solutions cover

All 43 questions in Inverse Trigonometric Functions are solved in the PDF. Here's what's inside, exercise by exercise:

Exercise 2.1

  1. Find the principal value of sin^-1(-1/2).
  2. Find the principal value of cos^-1(√3/2).
  3. Find the principal value of cosec^-1(2).
  4. Find the principal value of tan^-1(-√3).
  5. Find the principal value of cos^-1(-1/2).
  6. Find the principal value of tan^-1(-1).
  7. Find the principal value of sec^-1(2/√3).
  8. Find the principal value of cot^-1(√3).
  9. Find the principal value of cos^-1(-1/√2).
  10. Find the principal value of cosec^-1(-√2).
  11. Find the value of tan^-1(1) + cos^-1(-1/2) + sin^-1(-1/2).
  12. Find the value of cos^-1(1/2) + 2 sin^-1(1/2).
  13. If sin^-1 x = y, then (A) 0 ≤ y ≤ π (B) -π/2 ≤ y ≤ π/2 (C) 0 < y < π (D) -π/2 < y < π/2
  14. tan^-1(√3) - sec^-1(-2) is equal to (A) π (B) -π/3 (C) π/3 (D) 2π/3

Exercise 2.2

  1. Prove that 3 sin^-1 x = sin^-1(3x - 4x^3), x ∈ [-1/2, 1/2].
  2. Prove that 3 cos^-1 x = cos^-1(4x^3 - 3x), x ∈ [1/2, 1].
  3. Write tan^-1((√(1+x^2) - 1)/x), x ≠ 0 in the simplest form.
  4. Write tan^-1(√((1 - cos x)/(1 + cos x))), 0 < x < π in the simplest form.
  5. Write tan^-1((cos x - sin x)/(cos x + sin x)), -π/4 < x < 3π/4 in the simplest form.
  6. Write tan^-1(x / √(a^2 - x^2)), |x| < a in the simplest form.
  7. Write tan^-1((3a^2 x - x^3)/(a^3 - 3ax^2)), a > 0; -a/√3 < x < a/√3 in the simplest form.
  8. Find the value of tan^-1[2 cos(2 sin^-1(1/2))].
  9. Find the value of tan[(1/2)(sin^-1(2x/(1+x^2)) + cos^-1((1-y^2)/(1+y^2)))], |x| < 1, y > 0 and xy < 1.
  10. Find the value of sin^-1(sin(2π/3)).
  11. Find the value of tan^-1(tan(3π/4)).
  12. Find the value of tan(sin^-1(3/5) + cot^-1(3/2)).
  13. cos^-1(cos(7π/6)) is equal to (A) 7π/6 (B) 5π/6 (C) π/3 (D) π/6
  14. sin(π/3 - sin^-1(-1/2)) is equal to (A) 1/2 (B) 1/3 (C) 1/4 (D) 1
  15. tan^-1(√3) - cot^-1(-√3) is equal to (A) π (B) -π/2 (C) 0 (D) 2√3

Miscellaneous Exercise

  1. Find the value of cos^-1(cos(13π/6)).
  2. Find the value of tan^-1(tan(7π/6)).
  3. Prove that 2 sin^-1(3/5) = tan^-1(24/7).
  4. Prove that sin^-1(8/17) + sin^-1(3/5) = tan^-1(77/36).
  5. Prove that cos^-1(4/5) + cos^-1(12/13) = cos^-1(33/65).
  6. Prove that cos^-1(12/13) + sin^-1(3/5) = sin^-1(56/65).
  7. Prove that tan^-1(63/16) = sin^-1(5/13) + cos^-1(3/5).
  8. Prove that tan^-1(√x) = (1/2) cos^-1((1-x)/(1+x)), x ∈ [0, 1].
  9. Prove that cot^-1((√(1+sin x) + √(1-sin x))/(√(1+sin x) - √(1-sin x))) = x/2, x ∈ (0, π/4).
  10. Prove that tan^-1((√(1+x) - √(1-x))/(√(1+x) + √(1-x))) = π/4 - (1/2) cos^-1 x, -1/√2 ≤ x ≤ 1. [Hint: Put x = cos 2θ]
  11. Solve 2 tan^-1(cos x) = tan^-1(2 cosec x).
  12. Solve tan^-1((1-x)/(1+x)) = (1/2) tan^-1 x, (x > 0).
  13. sin(tan^-1 x), |x| < 1 is equal to (A) x/√(1-x^2) (B) 1/√(1-x^2) (C) 1/√(1+x^2) (D) x/√(1+x^2)
  14. sin^-1(1-x) - 2 sin^-1 x = π/2, then x is equal to (A) 0, 1/2 (B) 1, 1/2 (C) 0 (D) 1/2
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Read the Inverse Trigonometric Functions textbook chapter / PDF, or browse all CBSE Class 12 Mathematics solutions.

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